Claim replication ($V_T=X$)

A contingent claim is replicated when an admissible self-financing portfolio has exactly its terminal payoff. In the absence of arbitrage, two replicating strategies for the same payoff have the same initial cost.

With a constant cash account, holding $T-t+1$ shares during interval $(t-1,t]$ and selling one share at each endpoint replicates ${\sum }_{t=1}^T{S}_t$. Each sale is retained in cash; the initial cost is $T{S}_0$. The proof is a pathwise telescoping identity.

An admissible self-financing portfolio superhedges a contingent claim when its terminal wealth is at least the claim payoff almost surely. The least permitted initial cost is the superhedging price. Exact claim replication requires equality.

The infimum of initial costs of admissible superhedging portfolios. Positive pricing kernels supply lower bounds: if $\mathbb{E}[ZP]=p$ and $H\cdot P\ge X$, then $H\cdot p\ge \mathbb{E}[ZX]$ whenever these expectations exist.